Mean field portfolio games

Abstract

We study mean field portfolio games with random parameters, where each player is concerned with not only her own wealth, but also relative performance to her competitors. We use the martingale optimality principle approach to characterise the unique Nash equilibrium in terms of a mean field FBSDE with quadratic growth, which is solvable under a weak interaction assumption. Motivated by the latter, we establish an asymptotic expansion result in powers of the competition parameter. When the market parameters do not depend on the Brownian paths, we obtain the Nash equilibrium in closed form.

Appendices

Appendix A: $\theta$-dependent terms in the BSDE (3.12)

In this section, we summarise the cumbersome $\theta$-dependent terms in the BSDE (3.12). To facilitate the presentation, we introduce the notation

$$\begin{aligned} f^{\sigma}&= \frac{\sigma h}{(1-\gamma )(\sigma ^{2}+(\sigma ^{0})^{2} )},\qquad f^{ \sigma ^{0}}= \frac{\sigma ^{0} h}{(1-\gamma )(\sigma ^{2}+(\sigma ^{0})^{2} )}, \\ f^{h}&=\frac{ h^{2}}{(1-\gamma )(\sigma ^{2}+(\sigma ^{0})^{2} )}, \qquad \psi = \frac{ \sigma \sigma ^{0}}{ (1-\gamma )( \sigma ^{2}+(\sigma ^{0})^{2} ) }, \\ \psi ^{\sigma}&= \frac{\sigma ^{2}}{ (1-\gamma )( \sigma ^{2}+(\sigma ^{0})^{2} ) }, \qquad \psi ^{\sigma ^{0}}= \frac{(\sigma ^{0})^{2}}{ (1-\gamma )( \sigma ^{2}+(\sigma ^{0})^{2} ) }, \\ \phi ^{(1)}&= \frac{h\sigma ^{0}}{(1-\gamma )( \sigma ^{2}+(\sigma ^{0})^{2} )}+ \frac{\sigma \sigma ^{0}Z^{\bar{o}}}{(1-\gamma )( \sigma ^{2}+(\sigma ^{0})^{2} )} + \frac{(\sigma ^{0})^{2}Z^{0,\bar{o}}}{(1-\gamma )(\sigma ^{2}+(\sigma ^{0})^{2})}, \\ \phi ^{(2)}&= \frac{h\sigma}{(1-\gamma )( \sigma ^{2}+(\sigma ^{0})^{2} )}+ \frac{\sigma ^{2}Z^{\bar{o}}}{(1-\gamma )( \sigma ^{2}+(\sigma ^{0})^{2} )} + \frac{\sigma \sigma ^{0}Z^{0,\bar{o}}}{(1-\gamma )(\sigma ^{2}+(\sigma ^{0})^{2})}, \\ \phi ^{(3)}&=Z^{0,\bar{o}} + \frac{\gamma \sigma ^{0} (h+\sigma Z^{\bar{o}}+\sigma ^{0}Z^{0,\bar{o}} ) }{(1-\gamma )(\sigma ^{2}+(\sigma ^{0})^{2})}, \\ \phi ^{(4)}&= Z^{\bar{o}}+ \frac{\gamma \sigma ( h+\sigma Z^{\bar{o}}+\sigma ^{0}Z^{0,\bar{o}} ) }{(1-\gamma )(\sigma ^{2}+(\sigma ^{0})^{2})}, \\ \phi ^{\sigma}&=1+ \frac{\gamma \sigma ^{2}}{(1-\gamma )(\sigma ^{2}+(\sigma ^{0})^{2})}, \qquad \phi ^{\sigma ^{0}}=1+ \frac{\gamma (\sigma ^{0})^{2}}{(1-\gamma )(\sigma ^{2}+(\sigma ^{0})^{2})}, \\ g_{\cdot }&=-\mathbb{E}\bigg[ \frac{\theta \gamma (\sigma ^{0}_{\cdot})^{2} }{(1-\gamma )( \sigma ^{2}_{\cdot}+(\sigma ^{0}_{\cdot})^{2} ) } \bigg|\mathcal {F}^{0}_{\cdot }\bigg]. \end{aligned}$$

The terms that are dependent on $\theta$ are

$$ \mathcal {J}_{2}(t;\widetilde{Z},\widetilde{Z}^{0},\theta )=\mathcal {I}_{1}(t; \widetilde{Z})+\mathcal {I}_{2}(t;\widetilde{Z}^{0})+\mathcal {I}_{3}(t; \widetilde{Z},\widetilde{Z}^{0})+\mathcal {I}_{4}(t), $$

where the terms involving $\widetilde{Z}$ are given by

$$\begin{aligned} \mathcal {I}_{1}(t;\widetilde{Z}) &=\bigg( \frac{\phi ^{\sigma ^{0}}_{t}\theta ^{2}\gamma ^{2}}{2(1-g_{t})^{2}}+ \theta \gamma \mathbb{E}\bigg[ \frac{ \theta ^{2}\gamma ^{2} \psi ^{\sigma ^{0}}_{t} }{2(1-\gamma )(1-g_{t})^{2}} \bigg|\mathcal {F}^{0}_{t} \bigg]\bigg) \\ & \phantom{=:} \times ( \mathbb{E}[\psi _{t}\widetilde{Z}_{t}|\mathcal {F}^{0}_{t}] )^{2}+ \theta \gamma \mathbb{E}\bigg[ \frac{\psi ^{\sigma}_{t}}{2(1-\gamma )} ( \widetilde{Z}_{t})^{2} \bigg|\mathcal {F}^{0}_{t}\bigg] \\ & \phantom{=:} -\frac{\theta \gamma ^{2}\psi _{t}}{1-g_{t}}\widetilde{Z}_{t}\mathbb{E}[ \psi _{t}\widetilde{Z}_{t}|\mathcal {F}^{0}_{t}]-\theta \gamma \mathbb{E} \bigg[ \frac{\theta \gamma \psi _{t}}{(1-\gamma )(1-g_{t})} \widetilde{Z}_{t} \bigg|\mathcal {F}^{0}_{t} \bigg]\mathbb{E}[ \psi _{t} \widetilde{Z}_{t}|\mathcal {F}^{0}_{t} ] \\ & \phantom{=:} +\bigg( \frac{\theta \gamma}{1-g_{t}}\mathbb{E}[ \theta \gamma f^{ \sigma ^{0}}_{t}|\mathcal {F}^{0}_{t} ] -\frac{\theta \gamma}{1-g_{t}} \phi ^{(3)}_{t}\\ &\phantom{=:}\quad\ \ +\frac{\theta ^{2}\gamma ^{2}}{(1-g_{t})^{2}}\phi ^{ \sigma ^{0}}_{t}\mathbb{E}[ \phi ^{(1)}_{t}|\mathcal {F}^{0}_{t} ] \bigg) \mathbb{E}[ \psi _{t}\widetilde{Z}_{t}|\mathcal {F}^{0}_{t} ] \\ & \phantom{=:} +\theta \gamma \bigg( - \mathbb{E}\bigg[ \frac{\theta \gamma \phi _{t}^{(1)}}{(1-\gamma )(1-g_{t})}\bigg| \mathcal {F}^{0}_{t}\bigg] \\ & \phantom{=:} \qquad \quad + \mathbb{E}\bigg[ \frac{\theta ^{2}\gamma ^{2}\psi _{t}^{\sigma ^{0}}}{(1-\gamma )(1-g_{t})^{2}} \bigg|\mathcal {F}^{0}_{t}\bigg] \mathbb{E}[ \phi ^{(1)}_{t} |\mathcal {F}^{0}_{t} ] \bigg)\mathbb{E}[ \psi _{t}\widetilde{Z}_{t}|\mathcal {F}^{0}_{t} ] \\ & \phantom{=:} -\theta \gamma \mathbb{E}[ f^{\sigma}_{t}\widetilde{Z}_{t}|\mathcal {F}^{0}_{t} ]+\theta \gamma \mathbb{E}\bigg[ \frac{\phi _{t}^{(2)}}{1-\gamma} \widetilde{Z}_{t} \bigg|\mathcal {F}^{0}_{t} \bigg] \\ & \phantom{=:} -\theta \gamma \mathbb{E}[ \phi ^{(1)}_{t} |\mathcal {F}^{0}_{t}] \mathbb{E}\bigg[ \frac{ \theta \gamma \psi _{t}}{(1-\gamma )(1-g_{t})} \widetilde{Z}_{t} \bigg|\mathcal {F}^{0}_{t} \bigg]\\ & \phantom{=:}+\bigg(\phi _{t}^{(4)}- \frac{\theta \gamma ^{2}}{1-g_{t}}\psi _{t}\mathbb{E}[\phi ^{(1)}_{t}| \mathcal {F}^{0}_{t}] \bigg)\widetilde{Z}_{t}, \end{aligned}$$

the terms involving $\widetilde{Z}^{0}$ are given by

$$\begin{aligned} \mathcal {I}_{2}(t;\widetilde{Z}^{0}) &=\bigg( \frac{\phi _{t}^{\sigma ^{0}}\theta ^{2}\gamma ^{2}}{2(1-g_{t})^{2}}+ \theta \gamma \mathbb{E}\bigg[ \frac{\theta ^{2} \gamma ^{2}\psi _{t}^{\sigma ^{0}}}{2(1-\gamma )(1-g_{t})^{2}} \bigg|\mathcal {F}^{0}_{t} \bigg] \bigg) ( \mathbb{E}[\psi _{t}^{\sigma ^{0}} \widetilde{Z}^{0}_{t}|\mathcal {F}^{0}_{t} ] )^{2} \\ & \phantom{=:} -\frac{\theta \gamma \phi ^{\sigma ^{0}}_{t}}{1-g_{t}}\widetilde{Z}^{0}_{t} \mathbb{E}[ \psi ^{\sigma ^{0}}_{t}\widetilde{Z}^{0}_{t}|\mathcal {F}^{0}_{t} ] +\theta \gamma \mathbb{E}\bigg[ \frac{ \psi ^{\sigma ^{0}}_{t}}{2(1-\gamma )} (\widetilde{Z}^{0}_{t})^{2} \bigg|\mathcal {F}^{0} \bigg] \\ & \phantom{=:} -\theta \gamma \mathbb{E}\bigg[ \frac{\theta \gamma \psi _{t}^{\sigma ^{0}}}{(1-\gamma )(1-g_{t})} \widetilde{Z}^{0}_{t} \bigg|\mathcal {F}^{0}_{t} \bigg]\mathbb{E}[\psi _{t}^{ \sigma ^{0}}\widetilde{Z}^{0}_{t}|\mathcal {F}^{0}_{t} ] \\ & \phantom{=:} + \frac{\theta \gamma}{1-g_{t}}\bigg(\mathbb{E}[ \theta \gamma f^{ \sigma ^{0}}_{t}|\mathcal {F}^{0}_{t} ] -\phi ^{(3)}_{t} + \frac{\theta \gamma}{1-g_{t}}\phi ^{\sigma ^{0}}_{t}\mathbb{E}[ \phi ^{(1)}_{t}| \mathcal {F}^{0}_{t} ] \bigg)\mathbb{E}[ \psi ^{\sigma ^{0}}_{t} \widetilde{Z}^{0}_{t}|\mathcal {F}^{0}_{t} ] \\ & \phantom{=:} +\theta \gamma \bigg( - \mathbb{E}\bigg[ \frac{\theta \gamma \phi ^{(1)}_{t} }{(1-\gamma )(1-g_{t})} \bigg| \mathcal {F}^{0}_{t} \bigg] \\ & \phantom{=:} \quad \qquad +\mathbb{E}\bigg[ \frac{ \theta ^{2} \gamma ^{2}\psi ^{\sigma ^{0}}_{t} }{(1-\gamma )(1-g_{t})^{2}} \bigg|\mathcal {F}^{0}_{t}\bigg]\mathbb{E}[\phi ^{(1)}_{t}|\mathcal {F}^{0}_{t}] \bigg)\mathbb{E}[ \psi ^{\sigma ^{0}}_{t} \widetilde{Z}^{0}_{t}| \mathcal {F}^{0}_{t} ] \\ & \phantom{=:} +\theta \gamma \mathbb{E}\bigg[ \frac{ \phi _{t}^{(1)}}{1-\gamma} \widetilde{Z}^{0}_{t} \bigg|\mathcal {F}^{0}_{t} \bigg] -\theta \gamma \mathbb{E}[\phi ^{(1)}_{t}|\mathcal {F}^{0}_{t}] \mathbb{E}\bigg[ \frac{\theta \gamma \psi ^{\sigma ^{0}}_{t}}{(1-\gamma )(1-g_{t})} \widetilde{Z}^{0}_{t} \bigg|\mathcal {F}^{0}_{t} \bigg] \\ & \phantom{=:} -\theta \gamma \mathbb{E}[ f^{\sigma ^{0}}_{t}\widetilde{Z}^{0}_{t}| \mathcal {F}^{0}_{t} ]+\bigg(\phi ^{(3)}_{t}- \frac{\theta \gamma}{1-g_{t}}\phi ^{\sigma ^{0}}_{t}\mathbb{E}[\phi ^{(1)}_{t}| \mathcal {F}^{0}_{t}]\bigg) \widetilde{Z}^{0}_{t}, \end{aligned}$$

the cross-terms involving both $\widetilde{Z}$ and $\widetilde{Z}^{0}$ are given by

$$\begin{aligned} \mathcal {I}_{3}(t;\widetilde{Z},\widetilde{Z}^{0}_{t}) &= - \frac{\theta \gamma ^{2}\psi}{1-g_{t}}\widetilde{Z}_{t}\mathbb{E}[ \psi ^{\sigma ^{0}}_{t}\widetilde{Z}^{0}_{t}|\mathcal {F}^{0}_{t} ]- \frac{\theta \gamma \phi _{t}^{\sigma ^{0}}}{1-g_{t}}\mathbb{E}[ \psi _{t} \widetilde{Z}_{t}|\mathcal {F}^{0}_{t} ]\widetilde{Z}^{0}_{t} \\ & \phantom{=:} +\frac{\theta ^{2}\gamma ^{2}\phi _{t}^{\sigma ^{0}}}{(1-g_{t})^{2}} \mathbb{E}[\psi _{t}\widetilde{Z}_{t}|\mathcal {F}^{0}_{t}]\mathbb{E}[ \psi _{t}^{\sigma ^{0}}\widetilde{Z}^{0}_{t}|\mathcal {F}^{0}_{t} ] \\ & \phantom{=:} -\theta \gamma \mathbb{E}\bigg[ \frac{ \theta \gamma \psi _{t}^{\sigma ^{0}}}{(1-\gamma )(1-g_{t})} \widetilde{Z}^{0}_{t} \bigg|\mathcal {F}^{0}_{t} \bigg] \mathbb{E}[\psi _{t} \widetilde{Z}_{t}|\mathcal {F}^{0}_{t}] \\ & \phantom{=:} +\theta \gamma \mathbb{E}\bigg[ \frac{ \theta ^{2}\gamma ^{2}\psi _{t}^{\sigma ^{0}} }{(1-\gamma )(1-g_{t})^{2}} \bigg|\mathcal {F}^{0}_{t} \bigg] \mathbb{E}[ \psi _{t}\widetilde{Z}_{t}| \mathcal {F}^{0}_{t} ] \mathbb{E}[ \psi _{t}^{\sigma ^{0}}\widetilde{Z}^{0}_{t}| \mathcal {F}^{0}_{t} ] \\ & \phantom{=:} +\theta \gamma \mathbb{E}\bigg[ \frac{ \psi _{t}}{1-\gamma} \widetilde{Z}_{t}\widetilde{Z}^{0}_{t} \bigg|\mathcal {F}^{0}_{t} \bigg] \\ & \phantom{=:} -\theta \gamma \mathbb{E}\bigg[ \frac{ \theta \gamma \psi _{t} }{(1-\gamma )(1-g_{t})}\widetilde{Z}_{t} \bigg|\mathcal {F}^{0}_{t} \bigg] \mathbb{E} [ \psi _{t}^{\sigma ^{0}} \widetilde{Z}^{0}_{t}|\mathcal {F}^{0}_{t} ], \end{aligned}$$

and the remaining terms are given by

$$ \begin{aligned} \mathcal {I}_{4}(t) &= \frac{ (\theta \gamma )^{2}\phi ^{\sigma ^{0}}_{t} }{2(1-g_{t})^{2}}( \mathbb{E}[\phi ^{(1)}_{t}|\mathcal {F}^{0}_{t}])^{2} - \frac{\theta \gamma}{1-g_{t}} (- \mathbb{E}[ \theta \gamma f^{\sigma ^{0}}_{t} |\mathcal {F}^{0}_{t}]+\phi ^{(3)}_{t} )\mathbb{E}[\phi ^{(1)}_{t}| \mathcal {F}^{0}_{t}] \\ & \phantom{=:} +\theta \gamma \mathbb{E}\bigg[ \frac{(\sigma ^{2}_{t}+(\sigma ^{0}_{t})^{2})| \phi ^{(1)}_{t} |^{2} }{2} \bigg|\mathcal {F}^{0}_{t} \bigg] \\ & \phantom{=:} - \theta \gamma \mathbb{E}[ \phi ^{(1)}_{t}|\mathcal {F}^{0}_{t} ] \mathbb{E}\bigg[ \frac{\theta \gamma \phi _{t}^{(1)}}{(1-\gamma )(1-g_{t})} \bigg| \mathcal {F}^{0}_{t} \bigg] \\ & \phantom{=:} +\frac{\theta ^{3}\gamma ^{3}}{(1-g_{t})^{2}}\mathbb{E}\bigg[ \frac{\psi ^{\sigma ^{0}}_{t}}{2(1-\gamma )} \bigg|\mathcal {F}^{0}_{t} \bigg] ( \mathbb{E}[ \phi ^{(1)}_{t}|\mathcal {F}^{0}_{t} ] )^{2} \\ & \phantom{=:} -\theta \gamma \mathbb{E}[ f^{h}_{t}+f_{t}^{\sigma }Z^{\bar{o}}_{t} + f^{ \sigma ^{0}}_{t}Z^{0,\bar{o}}_{t} |\mathcal {F}^{0}_{t}]. \end{aligned} $$

Appendix B: Reverse Hölder inequality

We summarise the reverse Hölder inequality for a general stochastic process and for the stochastic exponential of a stochastic process in the space BMO, see Kazamaki [20, Theorem 3.1], which is used in the main text.

For some $p>1$, we say that a stochastic process $D$ satisfies the reverse Hölder inequality $R_{p}$ if there exists a constant $C$ such that for any $[0,T]$-valued stopping time $\tau$, it holds that

$$ \mathbb{E}\bigg[ \Big|\frac{D_{T}}{D_{\tau}} \Big|^{p}\bigg|\mathcal {F}_{ \tau }\bigg]\leq C. $$

Let $\Theta \in H^{2}_{{\mathrm{BMO}}}$ and $B$ be a Brownian motion. Define

$$ \mathcal {E}(\Theta )=\mathcal {E}\bigg( \int _{0}^{\cdot }\Theta _{s}\,dB_{s} \bigg). $$

Let $\Phi$ be the function defined on $(1,\infty )$ by

$$ \Phi (x)=\bigg( 1+\frac{1}{x^{2}}\log \frac{2x-1}{2(x-1)} \bigg)^{ \frac{1}{2}}-1. $$

Then $\Phi$ is continuous and decreasing with $\lim _{x\searrow 1}\Phi (x)=\infty$ and $\lim _{x\nearrow \infty}\Phi (x)=0$. Let $p_{\Theta}$ be the unique constant such that $\Phi (p_{\Theta})=\|\Theta \|_{{\mathrm{BMO}}}$. Then we have the following reverse Hölder inequality.

Lemma B.1

For any $1< p< p_{\Theta}$ and any stopping time $\tau$, $\mathcal {E}(\Theta )$ satisfies the reverse Hölder inequality $R_{p}$. In particular,

$$ \mathbb{E}\bigg[ \Big| \frac{\mathcal {E}_{T}(\Theta )}{\mathcal {E}_{\tau}(\Theta )} \Big|^{p} \bigg|\mathcal {F}_{\tau}\bigg]\leq K(p,\|\Theta \|_{{\mathrm{BMO}}}), $$

where

$$ K(p, x)=\frac{2}{1-\frac{2(p-1)}{2p-1}e^{p^{2}( x^{2}+2x )} }. $$

Appendix C: Energy inequality

We summarise here the energy inequality (see Kazamaki [20, Sect. 2.1] or Meyer [25, Remark 59, Chap. VII. 6]) in the form of Itô integrals, because this is frequently used in the main text.

Let $Z\in H^{2}_{{\mathrm{BMO}}}$ and $B$ be a Brownian motion. Then we have the following energy inequality for the Itô integral $\int _{0}^{\cdot }Z_{s}\,dB_{s}$.

Lemma C.1

For each integer $n$, it holds that

$$ \mathbb{E}\left [ \left \langle \int _{0}^{\cdot }Z_{s}\,dB_{s} \right \rangle ^{n}_{T} \right ] \leq n! \operatorname {ess\,sup}_{\tau}\left \| \mathbb{E}\left [ \left .\left \langle \int _{0}^{\cdot }Z_{s} \,dB_{s} \right \rangle _{T} - \left \langle \int _{0}^{\cdot }Z_{s} \,dB_{s} \right \rangle _{\tau }\right | \mathcal {F}_{\tau}\right ] \right \|_{ \infty}^{n}, $$

where the supremum is taken over all stopping times. Equivalently, for each integer $n$, we have

$$ \mathbb{E}\bigg[ \bigg( \int _{0}^{T} Z^{2}_{s}\,ds \bigg)^{n} \bigg] \leq n! \|Z\|^{2n}_{{\mathrm{BMO}}}. $$